Gallagher index

The Gallagher index "measures an electoral system’s relative disproportionality between votes received and seats allotted in a legislature."[1][2] As such, it measures the difference between the percentage of votes each party gets and the percentage of seats each party gets in the resulting legislature, and it also measures this disproportionality from all parties collectively in any one given election. That collective disproportionality from the election is given a precise score, which can then be used in comparing various levels of proportionality among various elections from various electoral systems.[3]

Michael Gallagher, who created the index, referred to it as a "least squares index", inspired by the sum of squared residuals used in the method of least squares. The index is therefore commonly abbreviated as "LSq" even though the measured allocation is not necessarily a least squares fit. The Gallagher index is computed by taking the square root of half the sum of the squares of the difference between percent of votes (Vi{\displaystyle V_{i}}) and percent of seats (Si{\displaystyle S_{i}}) for each of the political parties (i=1...n{\displaystyle i=1...n}).[4]

The index weighs the deviations by their own value, creating a responsive index, ranging from 0 to 100. The larger the differences between the percentage of the votes and the percentage of seats summed over all parties, the larger the Gallagher index. The larger the index value the larger the disproportionality and vice versa. Michael Gallagher included "other" parties as a whole category, and Arend Lijphart modified it, excluding those parties. Unlike the well-known Loosemore–Hanby index, the Gallagher index is less sensitive to small discrepancies.[6]

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The Gallagher index gained considerable attention in Canada in December 2016 in the context of efforts to reform Canada's electoral system.[7][8] The Special Committee on Electoral Reform (a Parliamentary Committee) recommended "that the Government should, as it develops
a new electoral system, use the Gallagher index in order to minimize the
level of distortion between the popular will of the electorate and the
resultant seat allocations in Parliament." The committee recommended that "the government should seek to
design a system that achieves a Gallagher score of 5 or less."[9][10] In the 2015 Canadian federal election, the Gallagher index was 12.02, where 0 would be a perfectly proportional election outcome.[11]

Thus the disproportionality of the 2005 New Zealand election is 1.13, which is very low by international standards.[13]

Note that the Māori Party has the highest difference, which is significantly above the others. This is due to New Zealand's system of reserved seats for Māori. The Māori seats are allocated by votes on a separate electoral roll, and while any party can contest these seats, they have historically been won by either the Māori Party, the Labour Party, or New Zealand First.

This table uses for example the 2012 Queensland state election, one of the largest landslides in Australian electoral history. Though Australia and New Zealand have somewhat similar political histories, Australia uses preferential voting in a First-Past-the-Post system, which tends to result in far less proportionality compared to New Zealand's MMP system (or other proportional electoral systems), especially for larger minor parties, such as The Greens or, historically, the Australian Democrats. The 2012 Queensland election had an extremely high Gallagher Index, at 31.16, due to the massive landslide in seats for the victorious LNP; most recent Australian state and federal elections however score between 10 and 12.

^Special Committee on Electoral Reform (a Canadian Parliamentary Committee) (December 1, 2016). Report 3: Strengthening Democracy in Canada : Principles, Process and Public Engagement for Electoral Reform (Report). Parliament of Canada. p. 69 (or p. 83 in PDF search). Retrieved December 26, 2016. One tool that has been developed to measure an electoral system's relative disproportionality between votes received and seats allotted in a legislature is the Gallagher Index, which was developed by Michael Gallagher (who appeared before the Committee).

^"Is Canada Fair?". Measuring Unfairness — Calculating Canada's Gallagher Index. (This website includes the Gallagher Index in adjustable table format. It initially shows the data for Canada's 2015 federal election, but some variables in some table cells are adjustable by the visitor to the website, and then the rest of the table is automatically adjusted to reflect this visitor's new input.). Retrieved 10 December 2016.

^The rules for federal elections in Canada require that certain provinces always get a certain quantity of seats – on a province by province basis. If so, then Byron Weber Becker proposed that the Gallagher index for Canada ought to ALSO reflect that. In other words, the Gallagher data should be collected on a province by province basis; and the Gallagher score should be calculated on a province by province basis. Only after that is done, can we then add up all of those provincial scores and then average them out to get the true national "composite Gallagher index" score. If we do that, then the illustrated table calculation of 12 for Canada is incorrect. It should instead show a "composite Gallagher index" of 17.1. Byron Weber Becker developed this "composite" index. See citation here: Special Committee on Electoral Reform (a Canadian Parliamentary Committee) (December 1, 2016). Report 3: Strengthening Democracy in Canada : Principles, Process and Public Engagement for Electoral Reform (Report). Parliament of Canada. p. 69 (or p. 83 in PDF search). Retrieved December 26, 2016. ...Professor Becker developed the “Gallagher Index Composite” for the Committee’s study...